@inproceedings { Gutman20150828,
author = "Gutman A.E.",
title = "The problem of existence of nonclosed Archimedean cones",
booktitle = "Geometry Days in Novosibirsk -- 2015. International conference (Novosibirsk, August 26--29, 2015): Proceedings",
address = "Novosibirsk",
publisher = "Sobolev Institute of Mathematics SB RAS",
year = "2015",
pages = "91--92",
language = "russian",
annote = "There are certain results that essentially reduce the class of locally convex spaces in which all Archimedean cones are closed, but, in general, the problem of describing such spaces remains open. We present the main results which are obtained in the course of solving the problem."
}
@article { Gutman20150908,
author = "Gutman A.E. and Emelyanov E.Yu. and Matyukhin A.V.",
title = "Nonclosed Archimedean cones in locally convex spaces",
journal = "Vladikavk. Math. J.",
year = "2015",
volume = "17",
number = "3",
pages = "36--43",
doi = "10.23671/VNC.2017.3.7262",
language = "russian",
annote = "The problem is stated of describing the class of locally convex spaces which include nonclosed Archimedean cones. Certain results are presented in the course of solving the problem.",
keywords = "Archimedean ordered vector space, locally convex space, weak topology, cone, wedge"
}
@inproceedings { Gutman20150914,
author = "Gutman A.E. and Matyukhin A.V.",
howpublished = "Electronic",
title = "The problem of describing the locally convex spaces which include nonclosed Archimedean cones",
booktitle = "Actual Questions of Contemporary Science. International scientific conference (Moscow, September 14--15, 2015): Proceedings",
address = "Moscow",
publisher = "RusAlliance Owl",
year = "2015",
isbn = "5990722532, 9785990722538",
pages = "8--12",
language = "russian",
annote = "The problem is stated of describing the class of locally convex spaces which include nonclosed Archimedean cones. Certain results are presented which are obtained in the course of solving the problem.",
keywords = "locally convex space, ordered vector space, Archimedean cone"
}
@article { Gutman20150926,
author = "Gutman A.E. and Matyukhin A.V.",
howpublished = "Electronic",
title = "Topological vector spaces with nonclosed Archimedean cones",
journal = "Prospero",
year = "2015",
volume = "8",
number = "20",
pages = "62--64",
language = "russian",
annote = "The problem is stated of describing the class of topological vector spaces with nonclosed Archimedean cones. Some results are presented in the course of solving the problem.",
keywords = "topological vector space, ordered vector space, axiom of Archimedes, cone"
}
@inproceedings { Gutman20160921,
author = "Gutman A.E. and Matyukhin A.V.",
title = "Nonclosed Archimedean cones",
booktitle = "Geometry Days in Novosibirsk -- 2016. International conference (Novosibirsk, September 21--24, 2016): Proceedings",
address = "Novosibirsk",
publisher = "Institute of Mathematics",
year = "2016",
pages = "46--47",
language = "russian",
annote = "The notion of Archimedean convex set is introduced and clarified. The problem is considered of describing the class of topological vector spaces which include nonclosed Archimedean cones. The main results are presented which are obtained in the course of solving the problem and its variations with cones replaced by wedges and closedness, by sequential closedness."
}
@inproceedings { Gutman20161212,
author = "Gutman A.E. and Matyukhin A.V.",
title = "Nonclosed Archimedean cones",
booktitle = "Geometric Analysis and Control Theory. International conference (Novosibirsk, December, 8--12, 2016): Proceedings",
address = "Novosibirsk",
publisher = "Sobolev Institute of Mathematics SB RAS",
year = "2016",
pages = "42--44",
annote = "The notion of Archimedean convex set is introduced and studied. The problem is considered of describing the class of topological vector spaces which include nonclosed Archimedean cones. The main results are presented which are obtained when solving the problem and its variations with cones replaced by wedges and closedness, by sequential closedness."
}
@inproceedings { Gutman20170818,
author = "Gutman A.E. and Matyukhin A.V.",
title = "Archimedean cones and incoming directions",
booktitle = "Mathematics in the Modern World. International conference dedicated to the 60th anniversary of the Sobolev Institute of Mathematics (Novosibirsk, August 14--19, 2017): Proceedings",
address = "Novosibirsk",
publisher = "Institute of Mathematics",
year = "2017",
pages = "154",
language = "russian",
annote = "The notion is introduced and studied of incoming direction for a given convex set at a given point, a new criterion is provided for a wedge to be Archimedean, and the question is answered of when a wedge is included in a half-space."
}
@inproceedings { Gutman20180919,
author = "Gutman A.E.",
title = "Archimedean and directionally closed cones",
booktitle = "Geometry Days in Novosibirsk -- 2018. International conference (Novosibirsk, September 19--22, 2018): Proceedings",
address = "Novosibirsk",
publisher = "Institute of Mathematics",
year = "2018",
isbn = "978-5-86134-220-9",
pages = "15",
annote = "A criterion is provided for a wedge to be Archimedean, which is based on the notion of closed set along a direction."
}
@article { Gutman20190710,
author = "Gutman A.E. and Emelianenkov I.A.",
title = "Lexicographic structures on vector spaces",
journal = "Vladikavk. Math. J.",
year = "2019",
volume = "21",
number = "4",
pages = "42--55",
doi = "10.23671/VNC.2019.21.44621",
language = "russian",
annote = "Basic properties are described of the Archimedean equivalence and dominance in a totally ordered vector space. The notion of (pre)lexicographic structure on a vector space is introduced and studied. A lexicographic structure is a duality between vectors and points which makes it possible to represent an abstract ordered vector space as an isomorphic space of real-valued functions endowed with a lexicographic order. The notions of function and basic lexicographic structures are introduced. Interrelations are clarified between an ordered vector space and its function lexicographic representation. A new proof is presented for the theorem on isomorphic embedding of a totally ordered vector space into a lexicographically ordered space of real-valued functions with well-ordered supports. A criterion is obtained for denseness of a maximal cone with respect to the strongest locally convex topology. Basic maximal cones are described in terms of sets constituted by pairwise nonequivalent vectors. The class is characterized of vector spaces in which there exist nonbasic maximal cones.",
keywords = "maximal cone, dense cone, totally ordered vector space, Archimedean equivalence, Archimedean dominance, lexicographic order, Hamel basis, locally convex space"
}
@article { Gutman20230503,
author = "Gutman A.E. and Emelianenkov I.A.",
title = "Locally convex spaces with all Archimedean cones closed",
journal = "Sib. Matem. Zh.",
year = "2023",
volume = "64",
number = "5",
pages = "945--970",
doi = "10.33048/smzh.2023.64.505",
language = "russian",
annote = "We provide an exhaustive description of the class of locally convex spaces in which all Archimedean cones are closed. We introduce the notion of quasidense set and prove that the above class consists of all finite-dimensional and countable-dimensional spaces $X$ whose topological dual $X'$ is quasidense in the algebraic dual $X^\#$ of $X$.",
keywords = "Archimedean ordered vector space, locally convex space, weak topology, cone, wedge"
}
@article { Gutman20230504,
author = "Gutman A.E. and Emelianenkov I.A.",
title = "Locally convex spaces with all Archimedean cones closed",
journal = "Sib. Math. J.",
year = "2023",
volume = "64",
number = "5",
pages = "1117--1136",
doi = "10.1134/S0037446623050051",
annote = "We provide an exhaustive description of the class of locally convex spaces in which all Archimedean cones are closed. We introduce the notion of quasidense set and prove that the above class consists of all finite-dimensional and countable-dimensional spaces $X$ whose topological dual $X'$ is quasidense in the algebraic dual $X^\#$ of $X$.",
keywords = "Archimedean ordered vector space, locally convex space, weak topology, cone, wedge"
}
@article { Gutman20231224,
author = "Gutman A.E. and Emelianenkov I.A.",
title = "Quasidenseness in $R^N$ and projective parallelotopes",
journal = "Sib. Matem. Zh.",
year = "2024",
volume = "65",
number = "2",
pages = "258--276",
doi = "10.33048/smzh.2024.65.204",
language = "russian",
annote = "We establish two new criteria for the closedness of Archimedean cones in countable-dimensional locally convex spaces in terms of projective parallelotopes and projective automorphisms. We also answer some open questions about quasidenseness and quasi-interior.",
keywords = "Archimedean ordered vector space, locally convex space, weak topology, cone, quasi-interior, quasidense set"
}
@article { Gutman20231225,
author = "Gutman A.E. and Emelianenkov I.A.",
title = "Quasidenseness in $R^N$ and projective parallelotopes",
journal = "Sib. Math. J.",
year = "2024",
volume = "65",
number = "2",
pages = "265--278",
doi = "10.1134/S0037446624020034",
annote = "We establish two new criteria for the closedness of Archimedean cones in countable-dimensional locally convex spaces in terms of projective parallelotopes and projective automorphisms. We also answer some open questions about quasidenseness and quasi-interior.",
keywords = "Archimedean ordered vector space, locally convex space, weak topology, cone, quasi-interior, quasidense set"
}